3^1 + 3^1/2*which is different from the way 3^ 2/3 breaksdown; = 3rd-Root of 3 ^2

Algebraic Number Properties (mostly dealing with exponents and roots)

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Number property #1)

Any number raised to an even exponent you know the result is positive or zero; BUT you cannot know the sign of the number/variable

Number property #2)

When a number or variable is raised to an odd exponent, result can be positive, negative, or zeroex. X^3 , y^5 -- (-4)^3 = -64

Number property #3)

When a variable is raised to an odd exponent, the sign of the result determines the sign of the variable & will always be one solution

Number property #4)

When a number between 0&1 is squared; the result becomes smaller-For all other real numbers greater than 1 or less than 0,(-), square of that number becomes greaterex. (1/2)^2 = 1/4 ; (.4)^2 = .16 ; -(.4)^2 = -.16

Units Digits w/ exponents

There is a pattern for determining units digits for products & exponents.

Pattern Units Digits w/ exponents - "2" -

= 2, 4, 8, 6, 2,.....

Pattern Units Digits w/ exponents - "3" -

= 3, 9, 7, 1, 3,...

Pattern Units Digits w/ exponents - "4" -

= 4, 6, 4, 6,....

Pattern Units Digits w/ exponents - "5" -

= 5, 5, 5, 5,....

Pattern Units Digits w/ exponents - "6" -

= 6, 6, 6, 6,....

Pattern Units Digits w/ exponents - "7" -

= 7, 9, 3, 1, 7, 9....

Pattern Units Digits w/ exponents - "8" -

= 8, 4, 2, 6, 8, 4,....

Pattern Units Digits w/ exponents - "9" -

= 9, 1, 9, 1,.....

Root Number Property 1)

For even roots of all positive numbers, 2!!! solutions existOne positive & One negative - However when sign for square root on gmat is used, only asking for positive/ or principal- There is 1 solution for even root of 0- An even root of a (-) number is not real & not on GMAT

Root Number Property 2)

Odd roots of all real numbers, there is exactly 1 solutionSolution can be positive, negative, or 0

Root Number Property 3)

Taking the sqrt of a number between 0 & 1 results in a number greater than original.- For all (+) numbers greater than 1, the sqrt of that number will be less than originalex. sqrt of 1/4 = 1/2 & sqrt of 25 = 5

Avoid unecessary multiplying if a # will be used as/in a dividend later. -Remember to:Multiply every term within parentheses by number on outsideMultiply ever term within one set with every term in another(FOIL)

Key to Factoring -

Basically reversing parentheses rules; pull out common factors to form parentheses. - With fractions, Factor both numerator and denominator individually 1st. Then look for like termsDont try and form & force like terms

Linear Equation means -

All variables have an exponent of 1

Key to 1 variable equations -

Get the variable by itself by performing a series of operations (+, -, *, /). Always perform same operation on both sides

Key to multi-variable equations -

Generally will need (n-equations, for n # of variables)However for exponent of small set problems, don't require full lot of equations to matchTypically will need to express 1 variable using the other variables in an equation. Then plug into other equations until able to solve.

Simultaneously solving by Adding/Subtracting both sides of 2 or more equations

Solve by putting 2 of the equations together to eliminate a variable. KEY is to eliminate a variable so, manipulate one or both equations all the way across to do so.Then use that product to plug into other equations

Quadratic Equations -

An equation that contains a squared variable & can not be solved by combining like terms &/or isolating unknown termsMust be in: aX^2 + bX + c = 0

If you multiply or divide both sides by a negative number, the inequality flips

KEY!!! ex. ( X/Y > 3 = true? )

B/C you don't know the sign of y, you don't know if you need to flip inequality when multiplying both sides by Y. There is insufficient data to perform operation. You must know whether y(or any variable) is (+) or (-)

Inequality Fact - multiplying or dividing variables

You are not allowed to multiply or divide by a variable in an inequality, unless!!! you are sure of its sign

Inequality Fact - Subtracting?

DO NOT subtract inequalities to eliminate terms, b/c its the same as adding the 2nd equation after it has been multiplied by -1. Always try and add together

Inequality Fact - eliminating terms?

You are allowed to add 2 inequalities together to eliminate terms and solve for another term, AS LONG AS the sign's are pointing the same direction

Another way to write an algebraic expression w/ 1 variable; x^2 + 5 is f=f(x) = x^2 + 5-Think of the input of this function as "x" value & output of function as value defined by what x^2 + 5 yields when a value of x is filled in

Domain of a function -

is defined as the set of all allowable inputs for the function. Usually domain is set of all real numbers;However when a sqrt(x+5) is in the denominator of a function, the domain is limited to value of x>5 b/c you can't have a "0" or negative square root

For problems asking about multiple functions -

You start with the outer function, and where ever there is an X, you replace with f(x) & solve for it.

Function Applications - Sequences -

Domain of a sequence(allowable input values) consists of positive integers. Then 1st term of a sequence is the output when input is =1; 2nd term of sequence is output when input is =2....